"Although most of the city's 86,000 eighth graders were matched with a high school this year, every year thousands of students don't get in anywhere and it doesn't matter whether they have good grades, test scores and attendance records. They have to apply all over again, with a much more limited list of schools to choose from."

And the following paper uses the fact that the proportion of unmatched students doesn't go to zero as the school system gets large, so in a very large school system like NYC, the number of initially unmatched students won't be tiny. (That doesn't mean that allowing longer lists wouldn't help.)

The connection between two mentioned papers is not clear to me.The latter, strongly motivated by couple problems in the MEDICAL MATCHING, says the ratio of UNSTABLE matching (not UNMATCHED students) does not converge to 0 when there exists COMPLEMENTARITY in the participants' preferences. Most of SCHOOL CHOICE papers including the former assume SUBSTITUTABILITY, thereby free from these complementarity issues.

Professor Yasuda, thanks for your comment. The connection between the two papers isn't in the main results, but in a fact about matching in large markets that is an essential part of the proofs of the couples results. The critical fact is that, in large markets in which participants can only list a relatively small number of preferences, there will be a lot of unfilled positions (and unmatched students)...In the NYC case, these students are the ones who have to be asked to submit additional preference lists.

Dear Professor Roth: Many thanks for your kind reply.I have studied on school choice used in Tokyo with your coauthor Fuhito (and other young economists). We recently discovered that (i) the actual mechanism in Tokyo (with slight extension) has a nice ex-ante welfare property compared to Boston mechanism and DA, and (ii) this mechanism can be seen as a version of SCHOOL PROPOSING DA.We also noticed that the Tokyo mechanism looks similar to the new mechanism going to be used in Seattle.We would like to send you a draft once it's ready (if it would be OK for you).